We introduce the scale axis transform, a new skeletal shape representation for bounded open sets O ⊂ &Rdbl;d. The scale axis transform induces a family of skeletons that captures the important features of a shape in a scale-adaptive way and yields a hierarchy of successively simplified skeletons. Its definition is based on the medial axis transform and the simplification of the shape under multiplicative scaling: the s-scaled shape O
s is the union of the medial balls of O with radii scaled by a factor of s. The s-scale axis transform of O is the medial axis transform of O s, with radii scaled back by a factor of 1/s. We prove topological properties of the scale axis transform and we describe the evolution s &rarr O s by defining the multiplicative distance function to the shape and studying properties of the corresponding steepest ascent flow. All our theoretical results hold for any dimension. In addition, using a discrete approximation, we present several examples of two-dimensional scale axis transforms that illustrate the practical relevance of our new framework. © 2009 ACM.